We demonstrate that, starting with a simple fermion wave function, the steady mixed state of the evolution of a class of Lindbladians, and the ensemble created by strong local measurement of fermion density without postselection can be mapped to the “Gutzwiller projected” wave functions in the doubled Hilbert space—the representation of the density matrix through the Choi-Jamiołkowski isomorphism. A Gutzwiller projection is a broadly used approach of constructing spin liquid states. For example, if one starts with a gapless free Dirac fermion pure quantum state, the constructed mixed state corresponds to an algebraic spin liquid in the doubled Hilbert space. We also predict that for some initial fermion wave function, the mixed state created following the procedure described above is expected to have a spontaneous “strong-to-weak” U(1) symmetry breaking, which corresponds to the emergence of superconductivity in the doubled Hilbert space. We also design the experimental protocol to construct the desired physics of mixed states.

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• Open quantum systems & decoherence
• Quantum spin liquid